1. Technical Field
This invention relates generally to the field of computer-implemented educational tools in mathematics. More specifically, this invention relates to computer-implemented educational tools for facilitating the teaching of mathematics by employing inventive approaches and techniques that support the teacher-student collaboration desired for a student mastering an area of mathematics.
2. Description of the Related Art
Mathematics may be difficult to each and may be difficult for students to learn. Fractions, for example, have been found to be both difficult to teach and difficult for students to learn. At the same time, fractions, as well as other areas of mathematics, are a pivotal topic in mathematics education. Strategic use of technology can help support the teacher-student collaboration required to master this wide-ranging subject area.
An example: The challenges with fractions.
“Learning about fractions in the upper elementary grades is hard. Really hard! Fractions are hard not only for children to learn but for teachers to teach.” This is how Marilyn Burns, the highly esteemed author on elementary mathematics education, begins her first of three extensive books on teaching fractions (Burns, 2001). The National Math Advisory Panel identified fractions as an area that requires special attention: “Difficulty with fractions (including decimals and percents) is pervasive and is a major obstacle to further progress in mathematics, including algebra” (National Math Advisory Panel, 2008, p. xix). This challenge is understandable. Fractions present major conceptual leaps for students.
Consider these factors:                Fractions can describe many different things. When Sarah drinks half of the water in the bottle, she is consuming part of a whole (½). When Jeremy eats three of nine carrot sticks, he is consuming part of a set ( 3/9 or ⅓). When Justin reads for 15 minutes, that represents ¼ of a common unit of time. When Hannah swims across the lake her feat is a measure of length (e.g., 1⅓ miles). And on it goes, from one quarter of a dollar, to ⅓ cup of flour, to ½ of an acre of land. Fractions represent so many different things!        Sophisticated reasoning is required to evaluate any fraction. Upon entering the topic of fractions, students must analyze the relationship between two numbers in order to understand a single value. For example, ⅛ is smaller than ¼, and ⅜ is larger than ¼.        The real value of fractions is dependent upon the unit, or whole, of which they are a part. ¾ is not always greater than ¼! ¾ of a county is a smaller region than ¼ of a continent.        Fractions present a plethora of new terms for students to master: numerator, denominator, equivalent, common, uncommon, proper, improper, and more.        Students must first learn what fractions mean, and then they must perform operations on these fractions. Some of these operations, like addition and subtraction with uncommon denominators, require multiple steps. Other operations, like multiplying and dividing fractions, seem very abstract to many people, and disconnected from anything in real life. (Can you find a real life example of ⅛÷ ⅓? It is possible, but certainly not trivial!)        
It would be advantageous to provide computer-implemented educational tools that address and target the particular challenges for both the student and teacher about the teaching of and the learning of particular areas of mathematics, e.g. fractions, as described hereinabove.